Commutation Theorem - Hilbert Algebras - Examples

Examples

• The Hilbert-Schmidt operators on an infinite-dimensional Hilbert space form a Hilbert algebra with inner product (a,b) = Tr (b*a).
• If (X, μ) is an infinite measure space, the algebra L∞ (X) L2(X) is a Hilbert algebra with the usual inner product from L2(X).
• If M is a von Neumann algebra with faithful semifinite trace τ, then the *-subalgebra M₀ defined above is a Hilbert algebra with inner product (a, b) = τ(b*a).
• If G is a unimodular locally compact group, the convolution algebra L1(G)L2(G) is a Hilbert algebra with the usual inner product from L2(G).
• If (G, K) is a Gelfand pair, the convolution algebra L1(K\G/K)L2(K\G/K) is a Hilbert algebra with the usual inner product from L2(G); here Lp(K\G/K) denotes the closed subspace of K-biinvariant functions in Lp(G).
• Any dense *-subalgebra of a Hilbert algebra is also a Hilbert algebra.