Approximate Unitary Equivalence Research Articles

Distance between unitary orbits in C∗-algebras with stable rank one and real rank zero

Let A be a C∗-algebra with stable rank one and real rank zero. In this paper, it is shown that the usual distance dU defined on the approximate unitary equivalence classes (or unitary orbits) of the positive elements in A is equal to the distance dW defined on morphisms from Cuntz semigroup of C0(0,1] to the Cuntz semigrout of A.

On Specht's Theorem in UHF C⁎-algebras

Specht's Theorem states that two matrices A and B in Mn(C) are unitarily equivalent if and only if tr(w(A,A⁎))=tr(w(B,B⁎)) for all words w(x,y) in two non-commuting variables x and y. In this article we examine to what extent this trace condition characterises approximate unitary equivalence in uniformly hyperfinite (UHF) C⁎-algebras. In particular, we show that given two elements a,b of the universal UHF-algebra Q which generate C⁎-algebras satisfying the UCT, they are approximately unitarily equivalent if and only if τ(w(a,a⁎))=τ(w(b,b⁎)) for all words w(x,y) in two non-commuting variables (where τ denotes the unique tracial state on Q), while there exist two elements a,b in the UHF-algebra M2∞ which fail to be approximately unitarily equivalent despite the fact that they satisfy the trace condition. We also examine a consequence of these results for ampliations of matrices.

The spatial Rokhlin property for actions of compact quantum groups

We introduce the spatial Rokhlin property for actions of coexact compact quantum groups on C⁎-algebras, generalizing the Rokhlin property for both actions of classical compact groups and finite quantum groups. Two key ingredients in our approach are the concept of sequentially split ⁎-homomorphisms, and the use of braided tensor products instead of ordinary tensor products.We show that various structure results carry over from the classical theory to this more general setting. In particular, we show that a number of C⁎-algebraic properties relevant to the classification program pass from the underlying C⁎-algebra of a Rokhlin action to both the crossed product and the fixed point algebra. Towards establishing a classification theory, we show that Rokhlin actions exhibit a rigidity property with respect to approximate unitary equivalence. Regarding duality theory, we introduce the notion of spatial approximate representability for actions of discrete quantum groups. The spatial Rokhlin property for actions of a coexact compact quantum group is shown to be dual to spatial approximate representability for actions of its dual discrete quantum group, and vice versa.

Convex sets associated to C⁎-algebras

For a separable unital C⁎-algebra A and a separable McDuff II1-factor M, we show that the space Homw(A,M) of weak approximate unitary equivalence classes of unital ⁎-homomorphisms A→M may be considered as a closed, bounded, convex subset of a separable Banach space – a variation on N. Brown's convex structure Hom(N,RU). Many separable unital C⁎-algebras, including all (separable unital) nuclear C⁎-algebras, have the property that for any McDuff II1-factor M, Homw(A,M) is affinely homeomorphic to the trace space of A. In general Homw(A,M) and the trace space of A do not share the same data (several examples are provided). We characterize extreme points of Homw(A,M) in many cases, and we give two different conditions – one necessary and the other sufficient – for extremality in general. The universality of C⁎(F∞) is reflected in the fact that for any unital separable A,Homw(A,M) may be embedded as a face in Homw(C⁎(F∞),M). We also extend Brown's construction to apply more generally to Hom(A,MU).

Approximate Unitary Equivalence of Finite Index Endomorphisms of AFD Factors

We characterize the condition for two finite index endomorphisms on an AFD factor to be approximately unitarily equivalent. The characterization is given by using the canonical extension of endomorphisms, which is introduced by Izumi. Our result is a generalization of the characterization of approximate innerness of endomorphisms of the AFD factors, obtained by Kawahiashi--Sutherland--Takesaki and Masuda--Tomatsu. Our proof, which does not depend on the types of factors, is based on recent development on the Rohlin property of flows on von Neumann algebras.

Unitary orbits in 1-dimensional NCCW complexes

Let A be a 1-dimensional NCCW complex. In this paper, it is shown that the usual distance d U defined on the approximate unitary equivalence classes (unitary orbits) of self-adjoints in A is equal to a distance d W defined on morphisms from the Cuntz semigroup of C 0 ( 0 , 1 ] to Cu ( A ) . In addition, another distance d P on approximately unitary equivalence classes related to spectral information of positive elements in the classes is proved to be equal to d U and d W on simple unital inductive limits of 1-dimensional NCCW complexes.

REMARKS ON -STABLE PROJECTIONLESS C*-ALGEBRAS

AbstractIt is shown that ${\mathcal{Z}}$-stable projectionless C*-algebras have the property that every element is a limit of products of two nilpotents. This is then used to classify the approximate unitary equivalence classes of positive elements in such C*-algebras using traces.

Approximation of complex symmetric operators

An operator \(T\) on a complex Hilbert space \(\mathcal {H}\) is called a complex symmetric operator if there exists a conjugate-linear, isometric involution \(C:\mathcal {H}\rightarrow \mathcal {H}\) so that \(CTC=T^*\). In this paper, we study the approximation of complex symmetric operators. By virtue of an intensive analysis of compact operators in singly generated \(C^*\)-algebras, we obtain a complete characterization of norm limits of complex symmetric operators and provide a classification of complex symmetric operators up to approximate unitary equivalence. This gives a general solution to the norm closure problem for complex symmetric operators. As an application, we provide a concrete description of partial isometries which are norm limits of complex symmetric operators.

Approximate unitary equivalence of normaloid type operators

In this paper, we explore approximate unitary equivalence of normaloid operators and classify several normaloid type operators including transaloid operators, polynomial-normaloid operators and von Neumann operators up to approximate unitary equivalence. As an application, we explore approximation of transaloid operators with closed numerical ranges. Among other things, it is proved that those transaloid operators with closed numerical ranges are norm dense in the class of transaloid operators.

Approximate Unitary Equivalence to Skew Symmetric Operators

An operator \(T\) on a complex Hilbert space \(\mathcal {H}\) is called skew symmetric if \(T\) can be represented as a skew symmetric matrix relative to some orthonormal basis for \(\mathcal {H}\). In this paper, we study the approximation of skew symmetric operators and provide a \(C^*\)-algebra approach to skew symmetric operators. We classify up to approximate unitary equivalence those skew symmetric operators \(T\in \mathcal {B(H)}\) satisfying \(C^*(T)\cap \mathcal {K(H)}=\{0\}\). This is used to characterize when a unilateral weighted shift with nonzero weights is approximately unitarily equivalent to a skew symmetric operator.

Approximate unitary equivalence in simple $C^{*}$-algebras of tracial rank one

Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple C*-algebra with tracial rank no more than one. Suppose that $\phi, \psi: C\to A$ are two unital monomorphisms. With some restriction on $C,$ we show that $\phi$ and $\psi$ are approximately unitarily equivalent if and only if [\phi]=[\psi] in KL(C,A) \tau\circ \phi=\tau\circ \psi for all tracial states of A and \phi^{\ddag}=\psi^{\ddag}, here \phi^{\ddag} and \psi^{\ddag} are homomorphisms from $U(C)/CU(C)\to U(A)/CU(A) induced by \phi and \psi, respectively, and where CU(C) and CU(A) are closures of the subgroup generated by commutators of the unitary groups of C and B.

Classification of homomorphisms into simple [formula omitted]-stable [formula omitted]-algebras

We classify unital monomorphisms into certain simple Z -stable C ⁎ -algebras up to approximate unitary equivalence. The domain algebra C is allowed to be any unital separable commutative C ⁎ -algebra, or any unital simple separable nuclear Z -stable C ⁎ -algebra satisfying the UCT such that C ⊗ B is of tracial rank zero for a UHF algebra B. The target algebra A is allowed to be any unital simple separable Z -stable C ⁎ -algebra such that A ⊗ B has tracial rank zero for a UHF algebra B, or any unital simple separable exact Z -stable C ⁎ -algebra whose projections separate traces and whose extremal traces are finitely many.

Unitary equivalences for essential extensions of $C^*$-algebras

Let A be a unital separable C * -algebra and B = C⊗K., where C is a unital C * -algebra. Let T: A → M(B)/B be a unital full essential extension of A by B. We show that there is a bijection between elements in a quotient group of Ko(B) onto the strong unitary equivalence classes of unital full essential extensions σ for which [σ] = [T] in KK 1 (A,B). Consequently, when this group is zero, unitarily equivalent full essential extensions are strongly unitarily equivalent. When B is a non-unital but σ-unital simple C * -algebra with continuous scale, we also study the problem when two approximately unitarily equivalent essential extensions are strongly approximately unitarily equivalent. A group is used to compute the strongly approximate unitary equivalence classes in the same approximate unitary equivalent class of essential extensions.

The range of approximate unitary equivalence classes of homomorphisms from AH-algebras

Let C be a unital AH-algebra and A be a unital simple C*-algebras with tracial rank zero. It has been shown that two unital monomorphisms $${\phi, \psi: C\to A}$$ are approximately unitarily equivalent if and only if $$ [\phi]=[\psi]\quad {\rm in}\quad KL(C,A)\quad {\rm and}\quad \tau\circ \phi=\tau\circ \psi \quad{\rm for\, all}\tau\in T(A),$$ where T(A) is the tracial state space of A. In this paper we prove the following: Given $${\kappa\in KL(C,A)}$$ with $${\kappa(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}$$ and with κ([1 C ]) = [1 A ] and a continuous affine map $${\lambda: T(A)\to T_{\mathfrak f}(C)}$$ which is compatible with κ, where $${T_{\mathfrak f}(C)}$$ is the convex set of all faithful tracial states, there exists a unital monomorphism $${\phi: C\to A}$$ such that $$[\phi]=\kappa\quad{\rm and}\quad \tau\circ \phi(c)=\lambda(\tau)(c)$$ for all $${c\in C_{s.a.}}$$ and $${\tau\in T(A).}$$ Denote by $${{\rm Mon}_{au}^e(C,A)}$$ the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map $$\Lambda: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++},$$ where KLT(C, A)++ is the set of compatible pairs of elements in KL(C, A)++ and continuous affine maps from T(A) to $${T_{\mathfrak f}(C).}$$ Moreover, we found that there are compact metric spaces X, unital simple AF-algebras A and $${\kappa\in KL(C(X), A)}$$ with $${\kappa(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}$$ for which there is no homomorphism h: C(X) → A so that [h] = κ.

On the classification problem for nuclear C∗-algebras

We exhibit a counterexample to Elliott’s classification conjecture for simple, separable, and nuclear C ∗ -algebras whose construction is elementary, and demonstrate the necessity of extremely fine invariants in distinguishing both approximate unitary equivalence classes of automorphisms of such algebras and isomorphism classes of the algebras themselves. The consequences for the program to classify nuclear C ∗ -algebras are far-reaching: one has, among other things, that existing results on the classification of simple, unital AH algebras via the Elliott invariant of K-theoretic data are the best possible, and that these cannot be improved by the addition of continuous homotopy invariant functors to the Elliott invariant.

Commutative ${C}^*$-subalgebras of simple stably finite ${C}^*$-algebras with real rank zero

Let X be a second countable, path connected, compact metric space and let A be a unital separable simple nuclear 2-stable real rank zero C*-algebra. We classify all the unital *-embeddings (up to approximate unitary equivalence) of C(X) into A. Specifically, we provide an existence and a uniqueness theorem for unital *-embeddings from C(X) into A.

Full extensions and approximate unitary equivalence

Full extensions and approximate unitary equivalence

Homomorphisms and strong approximate unitary equivalence

. For many classes of C*-algebras, two *-homomorphisms are approximately unitarily equivalent if and only if they induce the same element in KL. In this paper, we investigate what additional information are needed for approximate unitary equivalence to imply strong approximate unitary equivalence. We show there exists a K-theoretical obstruction.

Three Applications of the Cuntz Semigroup

Building on work of Elliott and coworkers, we present three ap- plications of the Cuntz semigroup: (i) for many simple C � -algebras, the Thomsen semigroup is recovered func- torially from the Elliott invariant, and this yields a new proof of Elliott's classification theorem for simple, unital AI algebras; (ii) for the algebras in (i), classification of their Hilbert modules is similar to the von Neumann algebra context; (iii) for the algebras in (i), approximate unitary equivalence of self-adjoint operators is characterised in terms of the Elliott invariant.

On the topology of the Kasparov groups and its applications

In this paper we establish a direct connection between stable approximate unitary equivalence for * -homomorphisms and the topology of the KK-groups which avoids entirely C * -algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this topology coincides with both Pimsner's topology and the Brown–Salinas topology. We study the generalized Rørdam group KL ( A , B ) = KK ( A , B ) / 0 ¯ , and prove that if a separable exact residually finite dimensional C * -algebra satisfies the universal coefficient theorem in KK-theory, then it embeds in the UHF algebra of type 2 ∞ . In particular such an embedding exists for the C * -algebra of a second countable amenable locally compact maximally almost periodic group.