Topological Rank Research Articles

Higher dimensional Yang–Mills theories and topological terms rank 2 integrable systems of Prym varieties

Higher dimensional Yang–Mills theories and topological terms rank 2 integrable systems of Prym varieties

Trivial Gleason parts and the topological stable rank of H ∞

We show that the subset of the maximal ideal space of H ∞ formed by the trivial Gleason parts is totally disconnected. A pair ( ƒ, g ) ∈ H ∞ × H ∞ is called a corona pair if [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /][inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] > 0, where [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] is the open unit disc. We prove that the set U 2 ( H ∞ ) of corona pairs is dense in H ∞ × H ∞ .

The algebra of almost periodic functions has infinite topological stable rank

The algebra of almost periodic functions has infinite topological stable rank

Reduction of topological stable rank in inductive limits ofC∗-algebras

We consider inductive limits A of sequences A 1 →A 2 →... of finite direct sums of C * -algebras of continuous functions from compact Hausdorff spaces into full matrix algebras. We prove that A has topological stable rank (tsr) one provided that A is simple and the sequence of the dimensions of the spectra of A i is bounded. For unital A, tsr(A)=1 means that the set of invertible elements is dense in A. If A is infinite dimensional, then the simplicity of A implies that the sizes of the involved matrices tend to infinity, so by general arguments one gets tsr(A i )≤2 for large enough is whence trsr(A)≤2. The reduction of tsr from two to one requires arguments which are strongly related to this special class of C * -algebras

Stable rank and approximation theorems in 𝐻^{∞}

It is conjectured that for H ∞ {H^\infty } the Bass stable rank ( bsr ) (\text {bsr}) is 1 1 and the topological stable rank ( tsr ) (\text {tsr}) is 2 2 . bsr ( H ∞ ) = 1 {\text {bsr}}({H^\infty })= 1 if and only if for every ( f 1 , f 2 ) ∈ H ∞ × H ∞ ({f_1},{f_2})\; \in {H^\infty } \times {H^\infty } which is a corona pair (i.e., there exist g 1 {g_{1}} , g 2 ∈ H ∞ {g_2} \in {H^\infty } such that f 1 g 1 + f 2 g 2 = 1 {f_1}{g_1} + {f_2}{g_2}= 1 ) there exists a g ∈ H ∞ g \in {H^\infty } such that f 1 + f 2 g ∈ ( H ∞ ) − 1 {f_1} + {f_2}g \in {({H^\infty })^{ - 1}} , the invertibles in H ∞ {H^\infty } ; however, it suffices to consider corona pairs ( f 1 , f 2 ) ({f_1},{f_2}) where f 1 {f_1} is a Blaschke product. It is also shown that there exists a g ∈ H ∞ g \in {H^\infty } such that f 1 + f 2 g ∈ exp ⁡ ( H ∞ ) {f_1} + {f_2}g \in \exp ({H^\infty }) if and only if log ⁡ f 1 \log {f_1} can be boundedly, analytically defined on { z ∈ D : | f 2 ( z ) | > δ } \{ {z \in \mathbb {D}:|\,{f_2}(z)| > \delta } \} , for some δ > 0 \delta > 0 . tsr ( H ∞ ) = 2 {\text {tsr}}({H^\infty })= 2 if and only if the corona pairs are uniformly dense in H ∞ × H ∞ {H^\infty } \times {H^\infty } ; however, it suffices to show that the corona pairs are uniformly dense in pairs of Blaschke products. This condition would be satisfied if the interpolating Blaschke products were uniformly dense in the Blaschke products. For b b an inner function, K = H 2 ⊖ b H 2 K= {H^2} \ominus b\,{H^2} is an H ∞ {H^\infty } -module via the compressed Toeplitz operators C f = P K T f | K {C_f}= {P_K}{T_f}{|_K} , for f ∈ H ∞ f \in {H^\infty } , where T f {T_f} is the Toeplitz operator T f g = f g {T_f}g= f\,g , for g ∈ H 2 g \in {H^2} . Some stable rank questions can be recast as lifting questions: for ( f i ) 1 n ⊂ H ∞ ({f_i})_1^n \subset {H^\infty } , there exist ( g i ) 1 n ({g_i})_1^n , ( h i ) 1 n ⊂ H ∞ ({h_i})_1^n \subset {H^\infty } such that ∑ i = 1 n ( f i + b g i ) h i = 1 \sum \nolimits _{i = 1}^n {({f_i} + b\,{g_i}){h_i}= 1} if and only if the compressed Toeplitz operators ( C f i ) 1 n ({C_{{f_i}}})_1^n may be lifted to Toeplitz operators ( T F i ) 1 n ({T_{{F_i}}})_1^n which generate B ( H 2 ) B({H^2}) as an ideal.

Invertibility and topological stable rank for semi-crossed product algebras

Invertibility and topological stable rank for semi-crossed product algebras

On the order of approximation in approximative triadic decompositions of tensors

Based on Alder's (1984) result on the equivalence of algebraic and topological border rank of tensors over algebraically closed ground fields, we give an upper bound for the order of approximation in approximative triadic decompositions. We also treat the analogous question for real closed fields. In particular we carry over Alder's result to the case of real closed fields.

Thin spectra and stable range conditions

The topological stable rank of a Banach algebra A is the least integer n = tsr( A) such that U n ( A) is dense in A n , where U n ( A) = {( a 1,…, a n ) ϵ A n : ∑ i = 1 n Aa i = A}. We give a spectral characterization of the condition “tsr( A) ⩽ n,” valid for arbitrary Banach algebras. For a commutative Banach algebra A, the condition tsr( A) ⩾ n is related to the presence of some kind of discs in the spectrum X( A).