Editing Maximal compact subgroup (section)

===Proof of uniqueness for semisimple groups===
{{harvtxt|Mostow|1955}} also related the general problem for semisimple groups to the case of GL(''n'', '''R'''). The corresponding symmetric space is the space of positive symmetric matrices. A direct proof of uniqueness relying on elementary properties of this space is given in {{harvtxt|Hilgert|Neeb|2012}}.

Let <math>\mathfrak{g}</math> be a real semisimple Lie algebra with [[Cartan involution]] σ. Thus the [[fixed point subgroup]] of σ is the maximal compact subgroup ''K'' and there is an eigenspace decomposition

:<math>\displaystyle{\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p},}</math>

where <math>\mathfrak{k}</math>, the Lie algebra of ''K'', is the +1 eigenspace. The Cartan decomposition gives

:<math>\displaystyle{G=K\cdot \exp \mathfrak{p} = K\cdot P = P\cdot K.}</math>

If ''B'' is the [[Killing form]] on <math>\mathfrak{g}</math> given by ''B''(''X'',''Y'') = Tr (ad X)(ad Y), then

:<math>\displaystyle{(X,Y)_\sigma=-B(X,\sigma(Y))}</math>

is a real inner product on <math>\mathfrak{g}</math>. Under the adjoint representation, ''K'' is the subgroup of ''G'' that preserves this inner product.

If ''H'' is another compact subgroup of ''G'', then averaging the inner product over ''H'' with respect to the Haar measure gives an inner product invariant under ''H''. The operators Ad ''p'' with ''p'' in ''P'' are positive symmetric operators. This new inner produst can be written as

:<math>(S\cdot X,Y)_\sigma,</math>

where ''S'' is a positive symmetric operator on <math>\mathfrak{g}</math> such that 
Ad(''h'')<sup>''t''</sup>''S'' Ad ''h'' = ''S'' for ''h'' in ''H'' (with the transposes computed with respect to the inner product). Moreover, for ''x'' in ''G'',

:<math>\displaystyle{\mathrm{Ad}\, \sigma(x)=(\mathrm{Ad}\,(x)^{-1})^t.}</math>

So for ''h'' in ''H'',

:<math>\displaystyle{S\circ \mathrm{Ad}(\sigma(h))= \mathrm{Ad}(h)\circ S.}</math>

For ''X'' in <math>\mathfrak{p}</math> define

:<math>\displaystyle{f(e^X)=\mathrm{Tr}\, \mathrm{Ad}(e^X) S.}</math>

If ''e''<sub>''i''</sub> is an orthonormal basis of eigenvectors for ''S'' with ''Se''<sub>''i''</sub> = λ<sub>''i''</sub> ''e''<sub>''i''</sub>, then

:<math>\displaystyle{f(e^X)=\sum \lambda_i (\mathrm{Ad}(e^X)e_i,e_i)_\sigma \ge (\min \lambda_i)\cdot \mathrm{Tr}\,e^{\mathrm{ad}\,X},}</math>

so that ''f'' is strictly positive and tends to ∞ as |''X''| tends to ∞. In fact this norm is equivalent to the operator norm on the symmetric operators ad ''X'' and each non-zero eigenvalue occurs with its negative, since i ad ''X'' is a ''skew-adjoint operator'' on the compact real form <math>\mathfrak{k}\oplus i\mathfrak{p}</math>.

So ''f'' has a global minimum at ''Y'' say. This minimum is unique, because if ''Z'' were another then

:<math>\displaystyle{e^Z=e^{Y/2} e^X e^{Y/2},}</math>

where ''X'' in <math>\mathfrak{p}</math> is defined by the Cartan decomposition

:<math>\displaystyle{e^{Z/2}e^{-Y/2}=k\cdot e^{X/2}.}</math>

If ''f''<sub>''i''</sub> is an orthonormal basis of eigenvectors of ad ''X'' with corresponding real eigenvalues μ<sub>''i''</sub>, then

:<math>\displaystyle{g(t)= f(e^{Y/2} e^{tX} e^{Y/2})= \sum e^{\mu_i t} \|Ad(e^{Y/2})f_i\|^2_\sigma.}</math>

Since the right hand side is a positive combination of exponentials, the real-valued function ''g'' is [[Strictly convex function|strictly convex]] if ''X'' ≠ 0, so has a unique minimum. On the other hand, it has local minima at ''t'' = 0 and ''t'' = 1, hence ''X'' = 0 and ''p'' = exp ''Y'' is the unique global minimum. By construction
''f''(''x'') = ''f''(σ(''h'')''xh''<sup>−1</sup>) for ''h'' in ''H'', so that ''p'' = σ(''h'')''ph''<sup>−1</sup> for ''h'' in ''H''. Hence σ(''h'')= ''php''<sup>−1</sup>. Consequently, if ''g'' = exp ''Y''/2, ''gHg''<sup>−1</sup> is fixed by σ and therefore lies in ''K''.