Let L be a subspace lattice which contains a sequence { P n } of commuting projections such that for any subsequence { P n i }, VP n i = I and ΛP n i = 0. Then Alg( L)∩ K( H)=0. Suppose G is a compact abelian group and E ⊆ G is a closed set. There is a family L E of commutative subspace lattices for which Alg( L E )∩ K( H)≠0 precisely when E is a set of multiplicity in the sense of harmonic analysis. By showing that the graph of ⩽ is a set of uniqueness in 2 ∞ × 2 ∞ we obtain a “thin set” proof that Alg(2 ∞, ⩽, m 1 2 ) contains no nonzero compact operators. For an important class of sets K ⊆ G × G A min( K) = A max( K) iff K is a set of spectral synthesis in the group G X G. It follows that if E 1and E 2 are sets of spectral synthesis in G then Alg(P E 1 \\ ̄ bo Alg (P E 2 ) = Alg(P E 1 ⊗ P E 2 iff E 1 × E 2 is a set of spectral synthesis in G X G and if semigroups ∑ 1 and ∑ 2 are sets of spectral synthesis in G then Alg(∑ 1) \\ ̄ bo Alg(∑ 2) = Alg(∑ 1 × ∑ 2 iff ∑ 1 × ∑ 2 is a set of spectral synthesis in G × G. The operator algebra Alg( P 1⊗ P 2 ⊗ …) is synthetic iff for all n, Alg( P 1 ⊗ … ⊗ P n is synthetic. This implies that the operator algebras Alg(2 ∞, ⩽, m p ) are synthetic.