By Simon Gindikin, James Lepowsky, Robert Wilson

These volumes include eighteen invited papers by way of special mathematicians in honor of the 80th birthday of Israel M. Gelfand, probably the most outstanding mathematicians of our time. Gelfand has performed an important function within the improvement of useful research over the past half-century. His paintings and his philosophy have in reality contributed to shaping our realizing of the time period 'functional analysis'. The papers in those volumes mostly quandary parts during which Gelfand has a truly powerful curiosity this present day, together with geometric quantum box thought, illustration concept, combinatorial buildings underlying quite a few 'continuous' buildings, quantum teams and geometry.

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**Sample text**

Reason: If this has been established, the truth of (b) will have been established with T replaced by U or |T |, and we could conclude that M T ξ, η = = = = M M M M U |T |ξ, η |T |ξ, U ∗ η ξ, |T |U ∗ η ξ, T ∗ η . ) In case T is unitary, the equation M ξ, η = M T ξ, T η is easily verified to define an assignment M ·, · which satisfies the requirements (i)-(iii) of an M -valued inner product. As for (iv), the assumed unitarity of T implies that if ξ, η ∈ X 0 , then trM ( M T ξ, T η ) = = T ξ, T η ξ, η and hence the uniqueness of the M -valued inner product now implies that M T ξ, T η = M ξ, η , from which the desired equality follows easily.

1). Then the unitarity axiom states that the function defined on [φ0 ]1 ∩ S(T ; B) by φ → d(φ1 ({σ(2), σ(4)}))d(φ1 ({σ(1), σ(3)})) Z(φ) yields a unitary matrix with rows and columns indexed by the ordered triples given by (φ2 (f1 (φ)), φ1 ({σ(2), σ(4)}), φ2 (f3 (φ))) and (φ2 (f4 (φ)), φ1 ({σ(1), σ(3)}), φ2 (f2 (φ))). 1. 3. Define the bases Bi for H(12)·φ (fi ((12) · φ)), in terms of the bases Bi chosen for the definition of the ‘U -matrix’, thus: B1 ((12) · φ) = B2 (φ) , B3 ((12) · φ) = F(12) (B3 (φ)), B2 ((12) · φ) = B1 (φ) , B4 ((12) · φ) = F(12) (B4 (φ)) .

B) Firstly, suppose T ∈ L(X, Y ) and has (left) polar decomposition T = U |T |. It is then true that T ∈ Hom(M X, M Y ) if and only if U ∈ Hom(M X, M Y ) and |T | ∈ End(M X). We omit the proof of this fact since it is exactly similar to the proof of the fact that a von Neumann algebra contains an operator if and only if it contains the ‘parts’ of the polar decomposition. ) With U as above, where T ∈ Hom(M X, M Y ), it should be clear that ran U and ran U ∗ are M -modules, and that U defines an M -linear unitary operator between these spaces.